Questions tagged [complex-dynamics]

This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.

Filter by
Sorted by
Tagged with
0 votes
0 answers
12 views

no wandering domain theorem for circles

I am trying to understand the proof of the no wandering domain theorem from Beardon's iterations of rational functions and thought a good start would be to omit the quasiconformal structures part and ...
user avatar
  • 625
2 votes
0 answers
42 views

Julia set of finite Blaschke product

I want to compute the Julia set of finite Blaschke product $B_3$ $$B_3 = (\frac{z+1/2}{z/2+1})^3$$ First I should analyze the map so I compute first derivative : $$ (3*(z+1/2)^2)/(z/2+1)^3-(3*(z+1/2)^...
user avatar
  • 1,554
2 votes
1 answer
51 views

Is the Mandelbrot set path-connected if and only if it is locally connected?

This question mentions that it's an open question whether the Mandelbrot set is path-connected and the answer conflated it with the more famous open question of whether the Mandelbrot set is locally ...
user avatar
3 votes
1 answer
79 views

What's the difference between a total basin of attraction and an immediate basin of attraction?

I understand that for an attracting fixed point $\hat{p}$ of a holomorphic self-map defined on some Riemann surface $S$ we define the total basin of attraction as $\mathcal{A}=\text{Bas}(\hat{p})=\{p\...
user avatar
  • 95
0 votes
0 answers
21 views

Prove a map is a covering map!

Let $a\in\mathbb{C}$ and $f:\mathbb{C} \to \mathbb{C}$ with $f(z)=z^3+a \bar{z}$. If $\Sigma$ denotes the critical set of $f$ then I want to know whether the restriction of $f$ to $\mathbb{C}\setminus\...
user avatar
0 votes
0 answers
29 views

Show $\partial R(A) \subseteq R(\partial A)$ for rational maps

For a nonconstant rational function $R: \mathbb{C}_\infty \longrightarrow \mathbb{C}_\infty$ I want to show that for any set $A$ it holds $$ \partial R(A) \subseteq R(\partial A). \\ $$ which, using $\...
user avatar
  • 625
0 votes
0 answers
14 views

Complex structure on parameter space $M_g$ of entire finite type map $g$

I was reading paper "Dynamical properties of some classes of entire functions" by A. Eremenko and M. Lyubich (http://www.numdam.org/item/?id=AIF_1992__42_4_989_0). My question is about ...
user avatar
  • 43
1 vote
0 answers
33 views

Finitely many Speiser graphs for a given entire holomorphic map of finite type?

Recently, I read definition of Speiser graph or, also called, line complex (see, for example there ). There is a certain ambiguity in its definition and I am going to formulate my question about it. ...
user avatar
  • 43
0 votes
1 answer
41 views

Julia set of $z^2+2z$ using conjugation of $z^2$

I wanted to calculate the Julia set of $S: z \mapsto z^2+2z$. I found that for $ \varphi: z \mapsto z-1 $ and $\varphi^{-1}: z \mapsto z+1$ the map $R: z \mapsto z^2$ is conjugate: $$ S(z) = z^2 + 2z =...
user avatar
  • 625
1 vote
1 answer
54 views

Is $ ds^2 = g_{11} dx^2 + 2g_{12}dxdy + g_{22}dy^2$ symmetric in the tangent vectors?

It's been a while since I studied differential geometry, so I forgot a lot of basic things. I got therefore stuck at a specific sentence in Milnor's book on complex dynamics. It says the following: A ...
user avatar
2 votes
1 answer
47 views

Multiplier of fixed point $\infty$ as limit

For $R: \mathbb{C}_\infty \rightarrow \mathbb{C}_\infty$ a rational map with $R(\infty) = \infty$ I know that the Multiplier of $\infty$ under R is: $$ m(R,\infty) = (\phi \ \circ \ R \ \circ \ \phi^{-...
user avatar
  • 625
0 votes
0 answers
49 views

Family $\lambda \exp(\exp(z))$

Consider family of entire holomorphic maps $E_\lambda(z) = \lambda \exp(\exp(z))$ for $\lambda \in \mathbb{C}^*$. Each of those maps doesn't have any critical values and has two finite asymptotic ones:...
user avatar
  • 43
2 votes
0 answers
49 views

Wandering domains of $z + \sin(2\pi z)$

I was recently working through Sullivan's Non-Wandering Theorem when I came across this counter-example to the theorem holding for functions $\mathbb{C} \to \mathbb{C}$. It appears that the entire ...
user avatar
2 votes
1 answer
77 views

Proof that Julia set is contained in closure of repelling periodic points

Kenneth Falconer's Fractal Geometry gives a proof that the Julia set of a polynomial $f$ is the closure of the repelling periodic points of $f$. To show that $J(f)$ is contained in the closure of the ...
user avatar
0 votes
1 answer
75 views

Are there known points on the boundary of the Mandelbrot set which iterate forever?

Where $f(z)=z^2+c$ is the Mandelbrot iteration function, are there any known complex numbers $z$ such that iterating $z\to f(z)$ to infinity retains $z$ on the boundary (i.e. it does not explode to ...
user avatar
  • 1,373
0 votes
0 answers
38 views

Are there "half step", or other subdivisions, of the Mandelbrot iteration function?

The Mandelbrot Set is generated by iterating $f(z)=z^2+c$. Is there a "half-step" function $g(z)$ such that $g(g(z))=f(z)$? What is the method for finding such half-iteration functions? ...
user avatar
  • 1,373
0 votes
0 answers
59 views

Julia Set Fixed Points And The Golden Ratio

When calculating fixed points for the Basilica Julia set ($z_{n+1}=z^2+c$, $c=-1$), the fixed points are given by $$ z^*_0(-1)=\frac{1 \mp \sqrt{1-4(-1)}}{2}=\frac{1 \mp \sqrt{5}}{2} $$ This is ...
user avatar
2 votes
0 answers
61 views

How to determine numerically where $c$ is in a mini-Mandelbrot set's filaments?

For example, how to tell efficiently which period $11$ island near a period $4$ island the coordinates $c \in \mathbb{C}$ correspond to, without tracing external rays? Ray tracing takes $O(N^2)$ time ...
user avatar
  • 4,789
0 votes
1 answer
47 views

Polynomials of a particular form

I have an iterative polynomial in fractal geometry, namely $Z = Z^2 + C$. What is the name of the polynomial of the more general form $Z = Z^\beta + Z^\gamma + ... + C$? I am calling them Julia-Fatou-...
user avatar
0 votes
1 answer
54 views

Explicitly calculating Green's function in complex dynamics

I was reading about Bottcher coordinates at infinity, and currently have a problem that would be most easily solved by calculating Green's function (more precisely, I would like to know whether ...
user avatar
  • 71
1 vote
1 answer
205 views

Mandelbrot set; are these trajectories chaotic?

I am using the complex Mandelbrot set, with an exponent of 2 so that the iterative equation is z = z^2 + c. The escape threshold is 4.0, and the maximum number of iterations is 5000. I find that all ...
user avatar
1 vote
1 answer
64 views

Automorphism classes of branched covers of disks over disks

I was reading Hubbard's book (Teichmuller Theory Vol 2) and in a proof (9.3.2), he mentions that there is one branched cover of the disk over a disk with 1 ramification point (degree $k$) up to ...
user avatar
  • 71
5 votes
0 answers
135 views

Why is there $\frac{9}{8}$ in this exponent?

The Mandelbrot set contains (countably) infinitely many baby Mandelbrot set copies. Each hyperbolic component (cardioid-like or disk-like shape) has a positive integer period $p$: the center of the ...
user avatar
  • 4,789
0 votes
1 answer
44 views

A question regarding the solution of a quadratic polynomial

I'd like to know why the following statement (taken from Iteration of Rational Functions by Beardon) is true: "Now $z^2+c$ has two fixed points, say $\alpha$ and $\beta$, in $\mathbb{C}$, and as ...
user avatar
  • 89
2 votes
1 answer
102 views

Asymptotic behavior of the quadratic recurrence $x_n=x_{n-1}^2+c$.

Let $c\in\mathbb{R}^+$. I am looking for a sequence $\{y_n\}$ that asymptotically (and if possible tightly) upper bounds the recurrence $$x_n=x^2_{n-1}+c.$$ I would to like to write $y_n$ as a ...
user avatar
  • 1,548
1 vote
0 answers
65 views

Generalized julia sets and simple connectedness

BACKGROUND: The Mandelbrot set is the set of complex numbers $c$ for which the sequence obtained by iterating $z \mapsto z^2 + c$ starting at $0$ remains bounded. For a complex number $c$, we can ...
user avatar
0 votes
0 answers
35 views

How does one use Green's theorem to do this conversion of integrals?

In Gamelin's book on Complex Dynamics, theorem 1.1 states that for $0 < r < 1$, we set $D_r = C - g(\Delta_r)$ where I used a - sign as opposed to backslash as a set difference, and the $g(z) = ...
user avatar
1 vote
1 answer
89 views

What are Bottcher coordinates useful for?

I've been reading about Bottcher coordinates lately, and I've seen them in fundamental books like Milnor's and Hubbard's. Although I can see the benefit of semiconjugating a map to a nicer map, I'm ...
user avatar
  • 71
0 votes
1 answer
46 views

Fault Recovery of a UAV using MATLAB

I'm currently investigating fault recovery in a UAV using MATLAB. I've been given several variables: phi = the roll angle psi = the yaw/heading angle beta = the side slip p = roll rate r = yaw/...
user avatar
  • 115
1 vote
0 answers
73 views

Radius of convergence of Taylor series related to Newton's method

Let $f$ be a degree $d$ polynomial, and $N(x)=x-f(x)/f'(x)$ be its Newton root-finding iteration. Suppose $f(c)=0$ and consider the Taylor series of $M_c(z)=N(c+z)-c$: what is its radius of ...
user avatar
  • 4,789
0 votes
0 answers
133 views

How does periodicity checking in the Mandelbrot Set work?

I was going through the Wikipedia page, on how periodicity checking works in the bulbs of the Mandelbrot Set. I do not understand why the code stores of a new x and ...
user avatar
2 votes
1 answer
89 views

What is the dimension of the "Julia set" generated by inverse iteration and why do I get numbers different from Hausdorff dimension

On the Julia set, the iterating the function $f:z\mapsto z^2+c$ generates a sequence of points on the Julia set. Because roundoff errors would be expected to cause the sequence to fall into an ...
user avatar
0 votes
1 answer
38 views

Why does mapping to unit disk and close to infinity imply non-normalcy of family of iterates?

I am reading Devaney's "Introduction to Chaotic Dynamical Systems", the chapter on the exponential family Prop. 9.1 I don't really understand how he arrives at the conclusion that if for $x\...
user avatar
  • 61
4 votes
0 answers
76 views

Determine the ray pair at the origin of a vein in the Mandelbrot set.

In a paper I found a description of subsets of the Mandelbrot set called veins (these are also described in earlier papers by other authors, for example Chapters 20-22 of the Orsay Notes by Adrien ...
user avatar
  • 4,789
1 vote
0 answers
52 views

Analytic Homeomorphism on unit circle

It's a problem from my complex dynamics lesson. Prove that $f:\mathbb{T}\to \mathbb{T}$ is a homoemorphism iff $\lvert a\rvert\geq 3$,where $\mathbb{T}$ is the unit circle and $$ f(z)=z^2\frac{z-a}{1-...
user avatar
  • 738
2 votes
0 answers
41 views

Homeomorphism of $\mathbb{S}^1$ of certain type

Suppose $f(z)=z^2\frac{z-a}{1-\overline{a}z}$, where $a\in\mathbb{C}$. Prove that $f:\mathbb{S}^1\to\mathbb{S}^1$ is a homeomorphism if and only if $|a|\geq3$. My idea: Surjectivity is quite evident. ...
user avatar
  • 193
4 votes
1 answer
176 views

Test to determine if a point is inside a cardioid whose cusp is not at the origin

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded within a circle of radius 2 when iterated from $z_0 = 0$. Plotted in the ...
user avatar
  • 751
4 votes
1 answer
108 views

Smooth Julia Set

My textbook tells me that for a polynomial $P$, its Julia set $J(P)$ is the nuit circle iff $P(z)=az^n$,where $\lvert a\rvert =1$,and $n\geq 2$. So I want to know whether there is a Julia set of a ...
user avatar
  • 738
1 vote
1 answer
68 views

Simplified expression for centers of period-three Mandlebrot bulbs

The Mandelbrot set contains three regions (two bulbs and a cardioid) with periods of three. These regions each contain a fixed point which is a root of the expression $x^3 + 2x^2 + x + 1$. The fixed ...
user avatar
  • 751
3 votes
1 answer
339 views

Finding attractors / fixed points for the circumference of the main bulb of the Mandelbrot Set

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded when iterated from $z_0 = 0$. Plotted in the complex plane it includes a main ...
user avatar
  • 751
6 votes
1 answer
142 views

How to estimate distance from root to nearest immediate basin boundary for Newton's method in one complex variable?

Context: I want to check that the atom domain size estimate is smaller than the inradius of the Newton immediate basin, for centers of hyperbolic components in the Mandelbrot set, and thus justify ...
user avatar
  • 4,789
1 vote
1 answer
30 views

Determine whether a point $c$ is in a wake $W_P$.

One can draw wakes in the parameter plane of the Mandelbrot set, by tracing external rays inwards from near $\infty$ (with Newton's method or other algorithm) to get polygonal outlines which can be ...
user avatar
  • 4,789
1 vote
1 answer
113 views

Attracting or parabolic cycles other than fixed points

I am studying the following complex polynomial $$P(z) = \frac{2z^{4}-2z^{3}+2z^2-z}{2z^{3}-2z^{2}+3z-2}$$ and I would like to know, if there are attracting or parabolic cycles for $P(z)$ different ...
user avatar
  • 125
2 votes
1 answer
81 views

Looking for an example of a function with a Julia set of positive Lebesgue measure.

I am looking for an example of a function with a Julia set of positive Lebesgue measure. I am certain these exist but cannot find a single example in the papers I have looked at. The function does not ...
user avatar
  • 346
1 vote
1 answer
54 views

Why is local connectivity important for polynomial Julia sets?

I'm trying to understand why local connectivity is important. I seem to remember a result that if the Julia set is locally connected then every external ray lands. I think this should mean we can get ...
user avatar
  • 33
1 vote
0 answers
80 views

A Quadratic Thurston Function Has Two Distinct Critical Points

My definitions are as follows. $\hat{\mathbb{C}}$ is the Riemann sphere. Degree of a continuous map $f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ is how many times it wraps around $\hat{\mathbb{C}}$...
user avatar
5 votes
2 answers
396 views

How can I reconstruct a Julia set by a given image?

Basically, I have two images of Julia sets I liked most from a google query $\ \ \ $ I want To be able to produce similar images, for that I need at least a palette from these images. To know the ...
user avatar
1 vote
0 answers
56 views

The values of the iteration $z^2 + c$ for c inside the Mandelbrot set

I'm wondering for which $c \in M$ (where M is the Mandelbrot set) the sequence $((z \to z^2 + c)^k(0))_k$ has a subsequence converging to $0$. One trivial solution is $c = 0$, though I cannot find any ...
user avatar
  • 123
0 votes
2 answers
242 views

Finding Periodic/Fixed Points in the Julia Sets close to the Period-3 Cardioid

The first image below shows the Julia Set at $-1.749512 + 0i$ (close to the base of the Period-3 Cardioid), and I'd like to find the periodic point located at where the white arrow is pointing at. ...
user avatar
7 votes
2 answers
218 views

Struggling to Understand Algorithm for Displaying Polynomial Matings with Julia Sets

I'm working on creating a program that visualizes projected Julia Sets on a Riemann Sphere (such as my video here) when I came across this website visualizing matings between Julia Sets, and I want to ...
user avatar

1
2 3 4 5
7